Abstract

We prove results about orbit closures and equidistribution for the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of the first two authors and a certain isolation property of closed $\mathrm{SL}(2,\mathbb{R})$ invariant manifolds developed in this paper.

[Mar2] G. A. Margulis, "Orbits of group actions and values of quadratic forms at integral points," in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part II, Jerusalem: Weizmann, 1990, vol. 3, pp. 127-150.

[Mar3] G. A. Margulis, "Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory," in Proceedings of the International Congress of Mathematicians, Vol. I, II, Tokyo, 1991, pp. 193-215.